Amyloid-β is an intrinsically disordered protein that forms fibrils in the brains of patients with Alzheimer's disease. To explore factors that affect the process of fibril growth, we computed the free energy associated with disordered amyloid-β monomers being added to growing amyloid fibrils using extensive molecular dynamics simulations coupled with umbrella sampling. We find that the mechanisms of Aβ40 and Aβ42 fibril elongation have many features in common, including the formation of an obligate on-pathway β-hairpin intermediate that hydrogen bonds to the fibril core. In addition, our data lead to new hypotheses for how fibrils may serve as secondary nucleation sites that can catalyze the formation of soluble oligomers, a finding in agreement with recent experimental observations. These data provide a detailed mechanistic description of amyloid-β fibril elongation and a structural link between the disordered free monomer and the growth of amyloid fibrils and soluble oligomers.